Methods of Evaluation of Limits

Let $a_1>a_2>a_3>\dots >a_n>1; p_1>p_2>p_3>\dots >p_n>0;$ such that $p_1+p_2+p_3+\dots +p_n=1$. Also $F\left(x\right)={\left(p_1{a}_1^x+p_2{a}_2^x+\dots +p_n{a}_n^x\right)}^{1/x},$ then $\underset{x\to \infty }{\lim }F\left(x\right)$ equals

If $\underset{x\to 0}{\lim }\frac{1-cosx.cos2x.cos3x\dots \dots \dots \cos nx}{x^2}$ has the value equal to$253$, find the value of $n$ is equal to (where $n\in N$)

The value of $\underset{x\to 0}{\lim }\frac{x^{6000}-{\left(\sin x\right)}^{6000}}{x^2{\left(\sin x\right)}^{6000}}$ is

The value of $\underset{x\to 0}{\lim }\frac{1+sinx-cosx+\ln \left(1-x\right)}{x·{tan}^{2}x}$ is

$\underset{x\to 0}{\lim }\frac{x+\ln \left(\sqrt{1+x^2}-x\right)}{x^3}=$

$\underset{x\to 0}{\lim }\frac{8}{x^8}\left[1-\cos \frac{x^2}{2}-\cos \frac{x^2}{4}+\cos \frac{x^2}{2}\cos \frac{x^2}{4}\right]$

Let n be an odd integer, if  $\sin n\theta ={\displaystyle \sum _{r=0}^n{b}_r{\sin }^r\theta }$ , for every value of  $\theta$ , then

The value of $\underset{n\to \infty }{\lim }{\left(1-\frac{1}{n}\right)}^{2n}$ is

Let $f\left(x\right)=\sin \frac{1}{x},x\ne 0$. Then $f\left(x\right)$ can be continuous at $x=0$

$\underset{x\to \pi /4}{\lim }\frac{1-\tan x}{1-\sqrt{2}\sin x}=$

If $f\left(9\right)=9$ and $f^{\text{'}}\left(9\right)=4$, then $\underset{x\to 9}{\lim }\frac{\sqrt{f\left(x\right)}-3}{\sqrt{x}-3}$ is equal to

$\underset{x\to \infty }{\lim }{\left(\frac{3x^2+2x+1}{x^2+x+2}\right)}^{\frac{6x+1}{3x+2}}$ is equal to

$\underset{x\to 0}{\lim }\left(\frac{a^x-b^x}{x}\right)=$

$\underset{x\to 0}{\lim } \frac{a^x-b^x}{c^x-d^x}$

If $S$ be the sum of coefficients in the expansion of ${\left(px+qy-rz\right)}^n$ (where $p, q, r>0$ ), then the value of $\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{S}{{\left(S^{\frac{1}{n}}+1\right)}^n},$ is:

Let $f$ and $g$ be two functions on $R$ defined by

$f\left(x\right)=\sqrt{x^2+1}-x$

$g\left(x\right)=\sin \left(\pi e^{1-x}\right)$

Define a function $h:R\to R$ by $h\left(x\right)=\max \left\{f\left(x\right),g\left(x\right)\right\}$. Then what can be said about $\underset{x\to \infty }{\lim }h\left(x\right)?$

$\underset{x\to 0}{\lim }\frac{\log \left(\sin 7x+\cos 7x\right)}{\sin 3x}$ equals

If $f\left(x\right)=\left|\begin{array}{ccc}\sin x& \cos x& \tan x\\ x^3& x^2& x\\ 2x& 1& 1\end{array}\right|$, then $\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x^2}$ is

$\underset{x\to 1}{\lim }\frac{4^x-4}{\sin \left(x-1\right)}=$

Evaluate:$\underset{x\to 0}{\lim }\frac{e^{ax}-1}{\sin bx}=$